A closed polyhedron cannot be made by snapping together seven equilateral triangles so that there are seven at every vertex, but if you persist and continue to snap together seven triangles at each vertex, you will actually construct an approximate model of the hyperbolic plane. It is this bumpy sheet with angular excesses all over the place that you might think of when you try to visualize the hyperbolic plane. Since we know that angular excess corresponds to negative curvature, we see that the hyperbolic plane is a negatively curved space. Hyperbolic geometry is also known as Non-Euclidean geometry...